The proof of Theorem REF 
is somewhat involved and forms the bulk of the paper.
We first show that it is sufficient to find a counterexample in the
homotopy category \(\mathbf {HoGpd}\)  of groupoids.
Then, given \(\kappa \)  as in the statement, we define a simple diagram \(D : \kappa \rightarrow \mathbf {HoGpd}\).
We make use of the theory of graphs of groups [1]} and the
associated fundamental groupoid [2]} in order to construct a
sufficiently pathological cocone \(D \rightarrow Z\)  which we use to show that
\(D\)  admits no \({\mathcal {G}}\) -privileged weak colimit.
This involves a detailed understanding of the morphisms in \(Z\)  and
how they are expressed as words in the given generators.
